Directional Derivative

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Directional Derivative

DEFINITION

The directional derivative of a Scalar Function

f(ec{x}) = f(x_1, x_2, \ldots, x_n)

along a vector

ec{v} = (v_1, \ldots, v_n)

is the Function defined by the Limit
:

D_{ec{v}}{f}(ec{x}) = \lim_{h 
ightarrow 0}{rac{f(ec{x} + hec{v}) - f(ec{x})}{h}}.

If the function

f

is differentiable at

ec{x}

, then the directional derivative exists along any vector

ec{v},

and one has

:

D_{ec{v}}{f}(ec{x}) = 
abla f(ec{x}) \cdot ec{v}

where

abla

denotes the Gradient and

\cdot

is the Euclidean Inner Product . At any point

p

, the directional derivative of

f

intuitively represents the Rate Of Change in

f

along

ec{v}

at the point

p

. Usually directions are taken to be normalized, so

ec{v}

is a Unit Vector , although the definition above works for arbitrary (even zero) vectors.

IN DIFFERENTIAL GEOMETRY

A Vector Field at a point

p

naturally gives rise to Linear Functional s defined on

p

by evaluating the directional derivative of a differentiable function

f

along the vector

ec{v}

where

ec{v}

is the vector of the Tangent Space at

p

assigned by the Vector Field . The value of the functional is then defined as the value of the corresponding directional derivative at

p

in the direction of

ec{v}

.

NORMAL DERIVATIVE

A normal derivative is a directional derivative taken in the direction normal (that is, Orthogonal ) to some surface in space, or more generally along a Normal Vector field orthogonal to some Hypersurface . See for example Neumann Boundary Condition .

SEE ALSO

Gradient

Partial Derivative

Lie Derivative

Differential Form

Gâteaux Derivative

Structure Tensor